Optimal. Leaf size=37 \[ \frac {8 \sqrt {4 x^2-9}}{243 x}+\frac {\sqrt {4 x^2-9}}{27 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {8 \sqrt {4 x^2-9}}{243 x}+\frac {\sqrt {4 x^2-9}}{27 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {-9+4 x^2}} \, dx &=\frac {\sqrt {-9+4 x^2}}{27 x^3}+\frac {8}{27} \int \frac {1}{x^2 \sqrt {-9+4 x^2}} \, dx\\ &=\frac {\sqrt {-9+4 x^2}}{27 x^3}+\frac {8 \sqrt {-9+4 x^2}}{243 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 25, normalized size = 0.68 \[ \frac {\sqrt {4 x^2-9} \left (8 x^2+9\right )}{243 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 28, normalized size = 0.76 \[ \frac {16 \, x^{3} + {\left (8 \, x^{2} + 9\right )} \sqrt {4 \, x^{2} - 9}}{243 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.13, size = 42, normalized size = 1.14 \[ \frac {32 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 3\right )}}{{\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 32, normalized size = 0.86 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (8 x^{2}+9\right )}{243 \sqrt {4 x^{2}-9}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.96, size = 29, normalized size = 0.78 \[ \frac {8 \, \sqrt {4 \, x^{2} - 9}}{243 \, x} + \frac {\sqrt {4 \, x^{2} - 9}}{27 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.87, size = 31, normalized size = 0.84 \[ \frac {8\,x^2\,\sqrt {4\,x^2-9}+9\,\sqrt {4\,x^2-9}}{243\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.35, size = 76, normalized size = 2.05 \[ \begin {cases} \frac {16 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{243} + \frac {2 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{27 x^{2}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {16 \sqrt {1 - \frac {9}{4 x^{2}}}}{243} + \frac {2 \sqrt {1 - \frac {9}{4 x^{2}}}}{27 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________